In Physics, a surface is a set of elements with pairwise measured distance d, satisfying the following properties:

  • For any two distinct elements A and B there are elements X and Y such that
    • d( Y, A ) < d( X, A ) < d( B, A ),
    • d( Y, X ) < d( X, A ),
    • d( Y, X ) < d( X, B ),
    • d( Y, B ) < d( X, B ),
    • d( Y, B ) < d( B, A ); and

  • for any four distinct elements A, B, C, and Q, which satisfy d( Q, A ) < d( C, B ) < d( C, A ) < d( B, A ) < d( C, A ) + d( C, B ), holds
    • either Vol( A, B, C, Q ) = 0, or
    • there exist elements J and K with d( K, A ) < d( J, A ) < d( C, A ) < d( J, A ) + d( K, A )
such that for all elements T with d( T, A ) < d( K, A ) holds
Vol( A, J, K, T ) / Area( A, J, K ) ≤ ½ Vol( A, B, C, Q ) / Area( A, B, C ).

Here Vol(), as a function of four elements, denotes the Volume of the corresponding 3-simplex (i.e. tetrahedron), expressed in terms of the six distance values (pairwise) between these four elements by Tartaglia's formula; and Area() , as a function of three elements, denotes the Area of the corresponding 2-simplex (i.e. triangle), expressed in terms of the three distance values (measured pairwise) between these three elements by Heron's formula.

In topology as applicable to physics, a surface is a topological space which satisfies for any three elements:

  • there are two closed sets with nonempty and disjoint interiorss, whose boundaries have these three elements in common, and
  • any third closed set with nonempty interior, whose border contains these three elements as well, shares some of its interior with either one or both of the first two closed sets. (Cmp. Kuratowski's theorem concerning the Graph K3, 3.)